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 Depdendent Variable

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 Dependent Variable

 Number of inequalities to solve: 23456789
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Objectives

– Recognize characteristics of parabolas
– Graph parabolas
– Determine a quadratic function’s minimum or
maximum value.

f(x)=ax2 +bx + c

graph to be a parabola.
Parabola

• Parabolas are
symmetric.

• Axis of Symmetry:
The line through
which the
parabola is
symmetric. Special Factored Form of the

• The vertex of the parabola is at (h,k)
and “a” describes the “steepness”
and direction of the parabola given by
the form

f (x) = a(x − h)2 + k
Minimum or maximum values of a function
occur at the VERTEX.

 P(x) = a(x – h)2 + k Vertex of parabola = (h, k)

a > 0 parabola opens up (h,k) = minimum point
Minimum Value of function is P(h)=k

a < 0 parabola opens down (h,k)=maximum point
Maximum Value of function is P(h)=k

Minimum (or maximum) function value for
a quadratic occurs at the vertex.

• If parabola opens up, f(x) has a
minimum value.

• If it opens down, f(x) has a maximum
value.

• Minimum/Maximum values are based on y-values.
Graph of f (x) = 2x2 − 4x + 3 = 2(x −1)2 Vertex Formula

P(x) = ax2 + bx + c (a ≠ 0)

The following formula will give you the x-value
for the vertex of a quadratic: Coordinates of vertex: Example

• Determine the following for f(x) without
graphing.
• f(x) = -3(x – 2)² + 12

a.) Find the vertex.
b.) Find the equation of the axis of symmetry.
c.) Does f(x) open up or down?
d.) Does f(x) have a max. or min. value?
Where does this value occur?
e.) What is the domain of f(x)?
f.) What is the range of f(x)?
Example

• Determine the following for f(x) without
graphing.
• f(x) = 2x² - 8 x - 3

a.) Find the vertex.
b.) Find the equation of the axis of symmetry.
c.) Does f(x) open up or down?
d.) Does f(x) have a max. or min. value?
Where does this value occur?
e.) What is the domain of f(x)?
f.) What is the range of f(x)?
To Graph a Quadratic Function

1. Find the coordinates of the vertex.
(Use the vertex formula.)

2. Determine which way parabola opens by looking at a.
a > 0 parabola opens up (Vertex is lowest point)
a < 0 parabola opens down (Vertex is highest point)

3. Find the x-intercept(s). (Set y = 0)

4. Find the y-intercept. (Set x = 0)

2. Graph additional points if needed by t-chart or symmetry.
Use the vertex and
intercepts to graph
f(x) = 5 – 4x – x²

Give the equation
of the axis of
symmetry.

Determine the
domain and the
range of f(x). 